Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438943 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35234 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,22,26,27)(15,24,17,21,25,28,16,20);(1,2)(3,4)(13,15,19,25)(14,20,22,21)(16,26,17,18)(23,28,27,24);(2,4)(6,9)(10,12)(13,16,18,15,19,17,26,25)(14,20,23,24,22,21,27,28);(2,4)(13,17,18,25,19,16,26,15)(14,21,23,28,22,20,27,24);(5,7,8,11)(6,10,9,12)(14,22)(20,21)(23,27)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=f,d^2=f,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6363678);